Motion and Ionization Equilibrium of Hydrogen Atoms in Superstrong Magnetic Field

Abstract

We study the effects of finite proton mass on the energy levels of hydrogen atoms moving transverse to a superstrong magnetic field $B$ with generalized pseudomomentum $K_\perp$. Field strengths of order $B\sim 10^{12}$ Gauss are typically found on the surfaces of neutron stars, but we also study the regime $B\go B_{crit}= 4.23\times 10^{13}$ Gauss, where the Landau excitation energy of the proton is large. We adopt two different approaches to the two-body problem in strong magnetic field, and obtain an approximate, but complete solution of the atomic energy as a function of $B$ and $K_\perp$. We show that, for $B>>B_{crit}$, there is an orthogonal set of bound states, which do not have any Landau excitation contribution in their energies. The states with very large $K_\perp$ have small binding energy and small transverse velocity, but are nevertheless distinct from the fully ionized states. The final results for the excitation energies are given in the form of analytical fitting formulae. The generalized Saha equation for the ionization-recombination equilibrium of hydrogen gas in the presence of a superstrong magnetic field is then derived. Although the maximum transverse velocity of a bound atom decreases as $B$ increases, the statistical weight due to transverse motion is actually increased by the strong magnetic field. For astrophysically interesting case of relatively low density and temperature, we obtain analytic approximations for the partition functions. The highly excited bound states have a smaller statistical weight than the fully ionized component.